Journal of Econometrics -...

Journal of Econometrics 146 (2008) 26–43

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Journal of Econometrics

journal homepage: www.elsevier.com/locate/jeconom

Forecasting the yield curve in a data-rich environment: A no-arbitragefactor-augmented VAR approachI

Emanuel Moench ∗Federal Reserve Bank of New York, 33 Liberty Street, New York, NY 10045, United States

a r t i c l e i n f o

Article history:Received 24 November 2005Received in revised form5 May 2007Accepted 27 June 2008Available online 6 July 2008

JEL classification:C13C32E43E44E52

Keywords:Yield curveFactor-augmented VARAffine term structure modelsDynamic factor modelsForecasting

a b s t r a c t

This paper suggests a term structure model which parsimoniously exploits a broad macroeconomicinformation set. The model uses the short rate and the common components of a large number ofmacroeconomic variables as factors. Precisely, the dynamics of the short rate are modeled with a Factor-Augmented Vector Autoregression and the term structure is derived using parameter restrictions impliedby no-arbitrage. The model has economic appeal and provides better out-of-sample yield forecastsat intermediate and long horizons than a number of previously suggested approaches. The forecastimprovement is highly significant and particularly pronounced for short and medium-term maturities.

Published by Elsevier B.V.

1. Introduction

Traditional models of the term structure decompose yieldsinto a set of latent factors. These models commonly provide agood in-sample fit to the data (e.g. Nelson and Siegel (1987),Knez et al. (1994) and Dai and Singleton (2000)) and can also beused to predict interest rates out-of-sample (e.g. Duffee (2002)and Diebold and Li (2006)). While providing a good statisticalfit, however, the economic meaning of such models is limitedsince they disregard the relationships between macroeconomicvariables and interest rates. In this paper, I suggest a model whichhas both economic appeal and superior predictive ability for yieldsas compared to traditional approaches.

In a widely recognized paper, Ang and Piazzesi (2003) augmenta standard three-factor affine term structure model with twomacroeconomic factors that enter the model through a Taylor-rule type of short rate equation. They find that the macro factors

I The views expressed in this article are those of the author and do not necessarilyreflect those of the Federal Reserve Bank of NewYork or the Federal Reserve System.∗ Tel.: +1 212 720 6625; fax: +1 212 720 1291.

E-mail address: [email protected].

0304-4076/$ – see front matter. Published by Elsevier B.V.doi:10.1016/j.jeconom.2008.06.002

account for a large share of the variation in interest rates and alsoimprove yield forecasts. Inspired by this finding, a vivid literaturehas emerged lately that explores different approaches to jointlymodel the term structure and the macroeconomy. Examples forsuch models are Hördahl et al. (2006), Diebold et al. (2006) andDewachter and Lyrio (2006).While these latter studies consistentlyfind thatmacroeconomic variables are useful for explaining and/orforecasting government bond yields, they only exploit very smallmacroeconomic information sets. Yet, by limiting the analysisto only a few variables, other potentially useful macroeconomicinformation is being neglected.1

This is particularly important for term structure modeling as arecent literature argues that the central bank acts in a ‘‘data-richenvironment’’ (Bernanke and Boivin, 2003). This means that themonetary policy authority bases its decisions upon a broad set ofconditioning information rather than only a few key aggregates.Consistent with this argument, a number of studies have foundthat factors which by construction summarize the comovement in

1 Note that the macroeconomic factors in Ang and Piazzesi (2003) are theprincipal components extracted from a group of four real and three nominalvariables, respectively. Accordingly, these authors employ a somewhat largermacroeconomic information set than the other studies referred to above.

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E. Moench / Journal of Econometrics 146 (2008) 26–43 27

a large number of macroeconomic time series help to explain andforecast the evolution of short-term interest rates (e.g. Bernankeand Boivin (2003), Giannone et al. (2004) and Favero et al. (2005)).In related work, Bernanke et al. (2005) suggest to combine theadvantages of factor modeling and structural VAR analysis byestimating a joint vector-autoregression of the short-term interestrate and factors extracted from a large cross-section of macrotime series. They label this approach a ‘‘Factor-Augmented VAR’’(FAVAR) and use it to analyze the dynamics of the short rate andthe effects of monetary policy on a wide range of macroeconomicvariables.

In this paper, I take the approach of Bernanke et al. (2005) astep further and employ the FAVAR model to study the dynamicsof the entire yield curve within an arbitrage-free model. Precisely,I suggest a model that has the following structure. A Factor-Augmented VAR is used to describe the dynamics of the short-terminterest rate conditional on a large macroeconomic informationset. Given the dynamics of the short rate, the term structure ofinterest rates is then derived using parameter-restrictions impliedby no-arbitrage. In sum, my model is an affine term structuremodel that has a Factor-Augmented VAR as the state equation, i.e.the short rate and the common components of a large number ofmacro time series represent the factors which drive the variationof yields. I label this approach a No-Arbitrage Factor-AugmentedVAR.

Estimation of the model is in two steps. First, I extract commonfactors from a large macroeconomic dataset using the methodsuggested by Stock and Watson (2002a,b) and estimate theparameters governing their joint dynamics with the monetarypolicy instrument in a VAR. Second, I estimate a no-arbitragevector autoregression of yields on the exogenous pricing factors.Specifically, I obtain the price of risk parameters byminimizing thesum of squared fitting errors of the model following the nonlinearleast squares approach of Ang et al. (2006). Altogether, estimationof the model is fast and it is thus particularly useful for recursiveout-of-sample forecasts.

The results of the paper can be summarized as follows. TheNo-Arbitrage FAVAR model based on four macro factors and theshort rate fits the US yield curvewell in-sample. More importantly,the model shows a strikingly good ability to predict yields out-of-sample. In a recursive out-of-sample forecast exercise, the No-Arbitrage FAVAR model is found to provide superior forecastswith respect to a number of benchmark models which havepreviously been suggested in the literature. Except for extremelyshort forecast horizons and very long maturities, the modelsignificantly outperforms the random walk, a standard three-factor affine model, the model suggested by Bernanke et al.(2004) which employs individual macroeconomic variables asfactors, and the model recently put forth by Diebold and Li(2006) which has been documented to be particularly useful forinterest rate predictions. A subsample analysis reveals that theNo-Arbitrage Factor-Augmented VAR model performs particularlywell in periods when interest rates vary a lot.

The paper is structured as follows. In Section 2, theNo-ArbitrageFactor-AugmentedVARmodel is presented and its parametrizationdiscussed. Section 3 describes the estimation of the model. InSection 4, I document the in-sample fit of the model and thendiscuss the results of the out-of-sample forecasts in Section 5.Section 6 concludes.

2. The model

Economists typically think of the economy as being affectedby monetary policy through the short term interest rate. Atthe same time, the central bank is often assumed to set theshort rate as a function of the overall state of the economy,

characterized e.g. by the deviations of inflation and outputfrom their desired levels. Bernanke et al. (2005) point out thattheoretical macroeconomic aggregates as output and inflationmight not be perfectly observable neither to the policy-makernor to the econometrician. Instead, they argue that the observedmacroeconomic time series should be thought of as noisymeasuresof economic concepts such as aggregate activity or inflation.Accordingly, these concepts should be treated as unobservable inempirical work so as to avoid confounding measurement error oridiosyncratic dynamics with fundamental economic shocks.

Bernanke et al. (2005) therefore suggest to extract a fewcommon factors from a large number of macroeconomic timeseries variables and to study the mutual dynamics of monetarypolicy and the key economic aggregates by estimating a jointVAR of the factors and the policy instrument, an approach whichthey label ‘‘Factor-Augmented VAR’’ (FAVAR). This approach canbe summarized by the following equations:

Xt = ΛF Ft + Λr rt + et (1)(Ftrt

)= µ + Φ(L)

(Ft−1rt−1

)+ ωt . (2)

Xt denotes aM × 1 vector of period-t observations of the observedmacroeconomic variables, ΛF and Λr are the M × k and M × 1matrices of factor loadings, rt denotes the short-term interest rate,Ft is the k × 1 vector of period-t observations of the commonfactors, and et is an M × 1 vector of idiosyncratic components.Moreover, µ = (µ′f , µr)

′ is a (k + 1) × 1 vector of constants, Φ(L)denotes the (k + 1) × (k + 1) matrix of order-p lag polynomialsand ωt is a (k + 1) × 1 vector of reduced form shocks withvariance covariance matrix Ω . Since affine term structure modelsare commonly formulated in state-space from, I rewrite the FAVARin Eq. (2) as

Zt = µ + ΦZt−1 + ωt , (3)

where Zt = (F ′t , rt , F′

t−1, rt−1, . . . , F′

t−p+1, rt−p+1)′, and where

µ, Φ, ω andΩ denote the companion form equivalents ofµ, Φ, ω,andΩ , respectively. Accordingly, the short rate rt can be expressedin terms of Zt as rt = δ′Zt where δ′ = (01×k, 1, 01×(k+1)(p−1)).

2.1. Adding the term structure

The term structure model which I suggest is built upon theidea that the Federal Reserve bases its decisions on a large set ofconditioning information and that the dynamics of the short-terminterest rate are therefore well described by a Factor-AugmentedVAR. Accordingly, yields are driven by the policy instrument aswell as the main shocks hitting the economy which are proxiedby the factors that capture the bulk of common variation in a largenumber of macroeconomic variables. I thus employ the FAVAR inEq. (3) as the state equation of my term structure model. To makethemodel consistentwith the assumption of no-arbitrage, I furtherimpose restrictions on the parameters governing the impact of thestate variables on the yields of different maturity. More precisely,I model the nominal pricing kernel as

Mt+1 = exp(

−rt −12λ′tΩλt − λ

′

tωt+1

),

= exp(

−δ′Zt −12λ′tΩλt − λ

′

tωt+1

), (4)

where λt are the market prices of risk. Following Duffee (2002),these are commonly assumed to be affine in the underlying statevariables Z , i.e.

λt = λ0 + λ1Zt . (5)

28 E. Moench / Journal of Econometrics 146 (2008) 26–43

In order to keep themodel parsimonious, I restrict the prices of riskto depend only on contemporaneous observations of the modelfactors. In an arbitrage-free market, the price of a n-months tomaturity zero-coupon bond in period t must equal the expecteddiscounted value of the price of an (n − 1)-months to maturitybond in period t + 1:

P (n)t = Et [Mt+1 P(n−1)t+1 ].

Assuming that yields are affine in the state variables, bond pricesP (n)t are exponential linear functions of the state vector:

P (n)t = exp(An + B′nZt

),

where the scalar An and the coefficient vector Bn depend on thetime to maturity n. Following Ang and Piazzesi (2003), I showin Appendix A that no-arbitrage is guaranteed by computingcoefficients An and Bn according to the following recursiveequations:

An = An−1 + B′n−1 (µ − Ωλ0) +12B′n−1ΩBn−1, (6)

B′n = B′

n−1 (Φ − Ωλ1) − δ′. (7)

Given the price of an n-months to maturity zero-coupon bond, thecorresponding yield is thus obtained as

y(n)t = −log P (n)t

n= an + b′nZt , (8)

where an = −An/n and b′n = −B′n/n.

Altogether, the suggestedmodel is completely characterized byEqs. (1), (3) and (6)–(8). In a nutshell, it is an essentially affineterm structure model that has a FAVAR as the state equation.Accordingly, I will refer to my model as a ‘‘No-Arbitrage Factor-Augmented VAR’’ approach.

3. Estimation of the model

In principle, the Factor-Augmented VAR model can be esti-mated using the Kalman filter and maximum likelihood. However,this approach becomes computationally infeasible when the num-ber of macro variables stacked in the vector X is large. Bernankeet al. (2005) therefore discuss two alternative estimationmethods:a single-step approach using Markov Chain Monte Carlo (MCMC)methods, and a two-step approach in which first principal compo-nents techniques are used to estimate the common factors F andthen the parameters governing the dynamics of the state equa-tion are obtained via standard classical methods for VARs. Com-paring both methods in the context of an analysis of the effects ofmonetary policy shocks, Bernanke et al. (2005) find that the two-step approach yields more plausible results. Another advantage ofthis method is its computational simplicity. Since recursive out-of-sample yield forecasts are the main focus of this paper, I thereforeemploy the principal components based approach in my applica-tion of the FAVAR model.

Accordingly, the common factors have to be extracted from thepanel of macro data prior to estimating the term structure model.As in Bernanke et al. (2005), this is achieved using standard staticprincipal components following the approach suggested by Stockand Watson (2002a,b). Precisely, let V denote the eigenvectorscorresponding to the k largest eigenvalues of the T × T cross-sectional variance–covariancematrix XX ′ of the data. Then, subjectto the normalization F ′F/T = Ik, estimates F̂ of the factors and Λ̂the factor loadings are given by

F̂ =√T V and

Λ̂ =√T X ′V ,

i.e. the common factors are estimated as the eigenvectorscorresponding to the k largest eigenvalues of the variance-covariance matrix XX ′.2 In practice, the true number of commonfactors which capture the common variation in the panel X is notknown. Bai and Ng (2002) have proposed some panel informationcriteriawhich allow to consistently estimate the number of factors.However, in the application of the FAVAR approach suggestedhere, the number of factors that can feasibly be included in themodel is limited due to computational constraints imposed by themarket prices of risk. I therefore fix the number of factors insteadof applying formal model selection criteria.

Given the factor estimates, estimation of the term structuremodel is performed using the consistent two-step approach of Anget al. (2006) which has also been employed in Bernanke et al.(2004). First, estimates of the parameters (µ, Φ, Ω) governing thedynamics of the model factors are obtained by running a VAR(p)on the estimated factors and the short term interest rate. Second,given the estimates from the first step, the parameters λ0 and λ1which drive the evolution of the state prices of risk, are estimatedby minimizing the sum of squared fitting errors of the model. Thatis, for a given set of parameter estimates (µ̂, Φ̂, Ω̂), the model-implied yields ŷ(n)t = ân + b̂′nZt are computed and the sum S isminimized with respect to λ0 and λ1 where S is given by3

S =T∑

t=1

N∑n=1

(ŷ(n)t − y(n)t )

2. (9)

Due to the recursive formulation of the bond pricing parameters,S is highly nonlinear in the underlying model parameters. It isthus helpful to find good starting values so as to achieve fastconvergence. This is done as follows. I first estimate the parametersλ0 assuming that risk premia are constant but nonzero, i.e. I set tozero all elements of the matrix λ1 which governs the time-varyingcomponent of the market prices of risk. I then take these estimatesofλ0 as starting values in a second step that allows for time-varyingmarket prices of risk, i.e. I let all elements of λ0 and λ1 be estimatedfreely.

This two-step approach potentially gives rise to an errors-in-variables bias since the estimation of the market price of riskparameters takes as given the estimated evolution of the states. Toadjust for this bias, I compute standard errors for λ0 and λ1 usinga Monte Carlo procedure which is described in Appendix B.

4. Empirical results

4.1. Data

I estimate the model using the following data. The macroeco-nomic factors are extracted from a dataset which contains about160monthly time series of various economic categories for the US.Among others, it includes a large number of time series related

2 To account for the fact that r is an observed factor which is assumedunconditionally orthogonal to the unobserved factors F in the model (1), its effecton the variables in X has to be isolated from the impact of the latent factors F . This isachieved by regressing all variables in X onto r and extracting principal componentsfrom the residuals of these regressions.

3 The assumption that only contemporaneous factor observations affectthe market prices of risk implies a set of zero restrictions on the pa-rameters λ0 and λ1 . In particular, λ0 = (λ̃′0, 01×(k+1)(p−1))

′ and λ1 =(λ̃1 0(k+1)×(k+1)(p−1)

0(k+1)(p−1)×(k+1) 0(k+1)(p−1)×(k+1)(p−1)

)where λ̃0 is of dimension (k + 1)

and λ̃1 is a (k+1)× (k+1)matrix. Hence, in practice only λ̃0 and λ̃1 need to be es-timated. Bernanke et al. (2004) impose the same set of restrictions in the estimationof their no-arbitrage macro VAR model.


to industrial production, more than 30 employment-related vari-ables, around 30 price indices and various monetary aggregates.It further contains different kinds of survey data, stock indices,exchange rates etc. This dataset has been compiled by Giannoneet al. (2004) to forecast US output, inflation, and short term inter-est rates. Notice that I exclude all interest rate related series fromthe original panel used by Giannone et al. (2004). The reason isthat if the factors of my arbitrage-free model were extracted froma dataset containing yields, restrictions would have to be imposedon the factor loading parameters in (1) so as to make them con-sistent with the assumption of no-arbitrage. This would imply anon-trivial complication of the estimation process. Accordingly, Iexclude the interest rate related series and thus implicitly assumethat the central bank does not take into account the informationcontained in yields when setting the short term rate. Notice alsothat this assumption implies that long-term interest rates do notaffect the evolution of the macroeconomy in my model.4

The principal components estimation of the common factors inlarge panels of time series requires stationarity. I therefore followGiannone et al. (2004) in applying different preadjustments to thetime series in the dataset.5 Finally, I standardize all series to havemean zero and unit variance.

I use data on zero-coupon bond yields of maturities 1, 3, 6, and9 months, as well as 1, 2, 3, 4, 5, 7, and 10 years. All interest ratesare continuously-compounded unsmoothed Fama–Bliss yields andhave been constructed from US treasury bonds using the methodoutlined in Bliss (1997). I estimate and forecast the model overthe post-Volcker disinflation period, i.e. from 1983:01 to the lastavailable observation of the macro dataset, 2003:09.

4.2. Model specification

In the first step of the estimation procedure, I extract commonfactors from the large panel of macroeconomic time series usingthe principal components approach of Stock andWatson (2002a,b).Together, the first 10 factors explain about 70% of the total varianceof all variables in the dataset. The largest contribution is accountedfor by the first four factors, however, which together explain about50% of the total variation in the panel. Table 1 lists the shares ofvariance explained by the first four factors aswell as the time seriesin the panel that each of them is most strongly correlated with.Note, however, that the factors estimated by principal componentsare only identified up to a non-singular rotation and therefore donot have a structural economic interpretation.

As already discussed above, the number of factors thatcan be included in the No-Arbitrage FAVAR model is limiteddue to parameterization constraints imposed by the marketprice of risk specification. Indeed, unless further restrictions areimposed on the market prices of risk, the number of parametersto estimate in the second step of the estimation procedureincreases quadratically with the number of factors. For the sakeof parsimony, I therefore restrict the number of factors to thefirst four principal components extracted from the large panelof monthly time series and the short rate. Unreported resultswith smaller and larger number of factors have shown thatthis specification seems to provide the best tradeoff betweenestimability and model fit. A similar choice has to be made

4 As discussed in Rudebusch et al. (2006a), this assumption is consistent withthe predictions of standard New-Keynesian models in which aggregate output isdetermined by a forward-looking IS curve and therefore only depends on expectedfuture short-term real interest rates.

5 Thoughwith a slight difference as regards the treatment of price series: insteadof computing first differences of quarterly growth rates as in Giannone et al. (2004),I follow Ang and Piazzesi (2003) and compute annual inflation rates.

Table 1Share of variance explained by factors and factor loadings

Factor 1 (25.1% of total variance) R2

Index of IP: Total 0.84Index of IP: Non-energy, total (NAICS) 0.84Index of IP: Mfg (SIC) 0.84Capacity utilization: Total (NAICS) 0.81Index of IP: Non-energy excl CCS (NAICS) 0.80

Factor 2 (10.9% of total variance)

CPI: All items less medical care 0.85CPI: Commodities 0.83CPI: All items (urban) 0.83CPI: All items less shelter 0.82CPI: All items ess food 0.79


CPI: Medical care 0.66PCE prices: Total excl food and energy 0.48PCE prices: Services 0.45M1 (in mil of current $) 0.39Loans and Securities @ all comm banks: Securities, U.S. govt (in mil of $) 0.37


Employment on nonag payrolls: Financial activities 0.27Employment on nonag payrolls: Other services 0.23Employment on nonag payrolls: Service-producing 0.19Employment on nonag payrolls: Mining 0.18Employment on nonag payrolls: Retail trade 0.17

This table summarizes R-squares of univariate regressions of the factors extractedfrom the panel of macro variables on all individual variables. For each factor, I listthe five variables that are most highly correlated with it. Notice that the series havebeen transformed to be stationary prior to extraction of the factors, i.e. for mostvariables the regressions correspond to regressions on growth rates. The four factorstogether explain about 50% of the total variation of the time series in the panel.

regarding the number of lags to include in the Factor-AugmentedVAR which represents the state equation of my term structuremodel. Applying the Hannan–Quinn information criterion with amaximum lag of 12 months indicates an optimal number of fourlags for the joint VAR of factors and the short rate. I thereforeemploy this particular specification for the in-sample estimationof the model. Note that in the recursive out-of-sample forecastexercise documented in Section 5, the lag length of the FAVAR isre-estimated each time a forecast is produced as it would have tobe in the context of truly real-time predictions.

4.3. Preliminary evidence

Before estimating the term structure model subject to no-arbitrage restrictions, I run a set of preliminary regressions tocheck whether the extracted macro factors are potentially usefulexplanatory variables in a term structure model. First, I usea simple encompassing test to assess whether a factor-basedpolicy reaction function provides a better explanation of monetarypolicy decisions than a standard Taylor-rule based on individualmeasures of output and inflation. I then perform unrestrictedregressions of yields on the model factors.

4.3.1. Do factors explain the short rate better than output andinflation?

The use of the Factor-Augmented VAR as a model for thedynamic evolution of short-term interest rates has been justifiedwith the argument that central banks base their monetarypolicy decisions on large sets of macroeconomic conditioninginformation rather than on individual measures of output andinflation alone. Whether this conjecture holds true empiricallycan be tested by comparing the fit of a standard Taylor-rulewith that of a policy reaction function based on dynamic factors.Bernanke and Boivin (2003) present evidence in favor of this


Table 2Policy rule based on individual variables

c ρ φy φπ

−0.011 0.955 1.332 2.592(0.078) (0.017) (0.627) (0.850)

This table reports estimates for a policy rule with partial adjustment based onindividual measures of output and inflation, i.e. rt = c + ρrt−1 + (1 − ρ)(φyyt +φππt ), where r denotes the federal funds rate, y the deviation of log GDP fromits trend, and π the annual rate of GDP inflation. The sample period is 1983:01 to2003:09. Standard errors are in parentheses. The R2 of this regression is 0.967.

Table 3Policy rule based on factors

c ρ φF1 φF2 φF3 φF4

0.198 0.957 0.115 0.076 −0.008 0.006(0.088) (0.016) (0.025) (0.031) (0.025) (0.026)

This table reports estimates for a policy rule with partial adjustment based on thefour factors extracted from a large panel of macroeconomic variables, i.e. rt =c + ρrt−1 + (1 − ρ)(φF1F1t + φF2F2t + φF3F3t + φF4F4t ), where r again denotesthe federal funds rate and F1 to F4 the four macro factors extracted from a panel ofabout 160monthly time series for the US. The sample period is 1983:01 to 2003:09.Standard errors are in parentheses. The R2 of this regression is 0.97.

claim by showing that the fitted value of the federal fundsrate from a factor-based policy reaction function is a significantadditional regressor in anotherwise standard Taylor-rule equation.Alternatively, one can separately estimate the two competingpolicy reaction functions and then perform an encompassing testà la Davidson and MacKinnon (1993). This is the strategy adoptedby Belviso and Milani (2005). I follow these authors and comparea standard Taylor rule with partial adjustment,6

rt = c + ρrt−1 + (1 − ρ)(φππt + φyyt),

to a policy reaction function based on the four factors whichrepresent state variables in the No-Arbitrage FAVAR model,

rt = c + ρrt−1 + (1 − ρ)φ′F Ft .

The results from both regressions are summarized in Tables 2 and3. As indicated by the regression R2s of 0.967 and 0.970, the factor-based policy rule fits the data slightly better than the standardTaylor rule.

The Davidson and MacKinnon (1993) encompassing test can beused to asses whether this improvement inmodel fit is statisticallysignificant. Accordingly, I regress the federal funds rate onto thefitted values from both alternative specifications which yields thefollowing result:

rt = 0.207r̂Taylort + 0.793 r̂

Factorst

= (0.186) (0.186).

Hence, the coefficient on the standard Taylor rule is insignif-icant whereas the coefficient on the factor-based fitted federalfunds rate is highly significant.7 This result can be interpreted asevidence supporting the hypothesis that the Fed reacts to a broadmacroeconomic information set.

6 Inflation π is defined as the annual growth rate of the GDP implicit pricedeflator (GDPDEF). The output gap is measured as the percentage deviation of logGDP (GDPC96) from its trend (computed using the Hodrick–Prescott filter and asmoothing parameter of 14400). Both quarterly series have been obtained fromthe St. Louis Fed website and interpolated to the monthly frequency using themethod described in Moench and Uhlig (2005). For the interpolation of GDP, I haveused industrial production (INDPRO), total civilian employment (CE16OV) and realdisposable income (DSPIC96) as related monthly series. CPI and PPI finished goodshave been employed as related series for interpolating the GDP deflator.

7 Unreported results have shown that this finding is robust to alternativespecifications of both reaction functions using a larger number of lags of the policyinstrument and the macro variables or factors.

Table 4Correlation of model factors and yields

y(1) y(6) y(12) y(36) y(60) y(120)

Panel A: Contemporaneous correlation of factors and yields

F1 0.243 0.318 0.351 0.382 0.389 0.379F2 0.597 0.619 0.617 0.570 0.546 0.537F3 0.150 0.153 0.161 0.270 0.340 0.407F4 0.315 0.325 0.331 0.354 0.365 0.380y(1) 1.000 0.987 0.975 0.936 0.899 0.833

Panel B: Correlation of 1 month lagged factors and yields

F1(−1) 0.296 0.365 0.393 0.409 0.409 0.393F2(−1) 0.600 0.614 0.610 0.564 0.539 0.531F3(−1) 0.145 0.152 0.161 0.269 0.342 0.411F4(−1) 0.296 0.309 0.316 0.346 0.358 0.373y(1)(−1) 0.984 0.974 0.960 0.923 0.888 0.822

Panel C: Correlation of 6 Months lagged factors and yields

F1(−6) 0.445 0.490 0.502 0.473 0.445 0.412F2(−6) 0.549 0.535 0.521 0.496 0.479 0.470F3(−6) 0.128 0.151 0.171 0.286 0.364 0.453F4(−6) 0.285 0.308 0.318 0.343 0.351 0.342y(1)(−6) 0.899 0.880 0.865 0.850 0.829 0.779

Panel D: Correlation of 12 months Lagged Factors and Yields

F1(−12) 0.548 0.567 0.564 0.502 0.455 0.390F2(−12) 0.448 0.405 0.385 0.398 0.400 0.408F3(−12) 0.145 0.186 0.205 0.303 0.378 0.479F4(−12) 0.275 0.309 0.329 0.349 0.354 0.348y(1)(−12) 0.742 0.712 0.705 0.738 0.745 0.723

This table summarizes the correlation patterns between the yields and factors usedfor estimating the term structuremodel. F1, F2, F3 and F4 denote themacro factorsextracted form the large panel of monthly economic time series for the US, y(1) toy(120) denote the yields of maturities 1-month to 10-years, respectively.

4.3.2. Unrestricted estimation of the term structure modelTo get a first impression whether the factors extracted from

the panel of macro variables also capture predictive informationabout interest rates of higher maturity, Table 4 summarizes thecorrelations between the yields and various lags of the factors oftheNo-Arbitrage FAVARmodel. This table shows that the short rateis most strongly correlated with yields of any other maturity. Yet,the fourmacro factors extracted from the panel ofmonthly US timeseries also exhibit considerable correlation with interest rates ofhigher maturity. While the short rate is contemporaneously moststrongly correlated with yields, the correlations between macrofactors and yields tend to be higher for longer lags. This indicatesthat the factors extracted from the panel of macro data might beuseful for forecasting interest rates.

To further explore the question whether the model factorshave explanatory power for yields, Table 5 provides estimates ofan unrestricted regression of yields of different maturities ontoa constant, the four macro factors and the 1-month Tbill, i.e. itestimates the pricing equation for yields,

Yt = A + BZt + ut ,

where no cross-equation restrictions are imposed on the coeffi-cients A and B. The first observation to make is that the R2 of theseregressions are all very high. Together with the short rate, the fourfactors explain more than 95% of the variation in short yields, andstill more than 85% of the variation in longer yields. Not surpris-ingly, the 1-month Tbill is the most highly significant explanatoryvariable for short maturity yields. However, in the presence of themacro factors its impact decreases sharply towards the long endof the maturity spectrum. This shows that the factors extractedfrom the large panel of macro variables exhibit strong explanatorypower for longer yields and thus represent potentially useful statevariables in a term structure model.


Table 5Unrestricted regressions of yields on factors

y(6) y(12) y(36) y(60) y(120)

cst 0.65 1.04 2.29 3.18 4.58[3.47] [3.58] [7.58] [10.65] [12.90]

F1 0.23 0.34 0.45 0.50 0.52[5.23] [4.83] [6.21] [6.93] [7.25]

F2 0.19 0.26 0.26 0.30 0.45[3.63] [2.81] [1.95] [2.12] [2.88]

F3 0.04 0.08 0.37 0.55 0.72[1.43] [1.82] [4.93] [6.32] [6.10]

F4 0.10 0.15 0.26 0.33 0.44[3.53] [3.01] [2.57] [2.75] [2.96]

y(1) 0.95 0.93 0.82 0.72 0.52[28.64] [17.59] [11.71] [9.07] [5.57]

R̄2 0.98 0.97 0.93 0.91 0.86

This table summarizes the results of an unrestricted VAR of yields of differentmaturities on the four macro factors extracted from the panel of economic timeseries, and the short rate. The estimation period is 1983:01 to 2003:09. t-values arein brackets.

4.4. Estimating the term structure model

4.4.1. In-sample fitIn this section, I report results obtained from estimating the

FAVAR model subject to the cross-equation restrictions (6) and(7) implied by the no-arbitrage assumption. The model fits thedata surprisingly well, given that it does not involve any latentyield curve factors as traditional affine models. Table 6 reports thefirst and second moments of observed and model-implied yields.These numbers show that on average the No-Arbitrage FAVARmodel provides a good fit to the yield curve. Fig. 1 provides avisualization of this result by showing the close match betweenaverage observed and model-implied yields across the entirematurity spectrum. Notice that the model seems to be missingsome of the variation in longer maturities since the standarddeviations of fitted interest rates are somewhat lower than thestandard deviations of the observed yields, especially at the longend of the curve. This can also be seen in Fig. 2 which plots thetime series for a selection of observed and model-implied yields.

Overall, the No-Arbitrage FAVAR model is able to capture thecross-sectional variation of government bond yields quite well,with a somewhat better in-sample fit at the short end of thecurve. As will be shown further below, this finding is paralleledby the prediction results obtained from the model. In particular,the forecast improvement of the No-Arbitrage FAVAR model overlatent yield factor based term structuremodels is found to bemorepronounced at the short than at the long end of the yield curve.

4.4.2. Parameter estimatesThe two-step procedure used to estimate the model implies a

potential errors-in-variables bias since it takes as given the state

Fig. 1. Observed and model implied average yield curve. This figure plots averageobserved yields against those implied by the No-Arbitrage FAVAR model.

evolution when estimating the market price of risk parameters. Toadjust for this bias, I compute standard errors for the market priceof risk parameters using a bootstrap procedure which is describedin Appendix B.

Table 7 reports the parameter estimates and associatedstandard errors of the No-Arbitrage FAVAR model. The firstpanel shows parameter estimates of the Factor-Augmented VARwhich represents the state equation of the model. The parameterestimates and corresponding standard errors have been obtainedby standard OLS procedures. A noticeable feature of the FAVARestimates is that most of the off-diagonal elements of the lags ofthe coefficient matrix Φ are insignificant. Hence, in addition tothe unconditional orthogonality of the model factors that followsfrom the estimation by principal components, there is also littleconditional correlation between the factors of the FAVAR model.

The second panel provides the estimates of the state prices ofrisk which constitute the remaining components of the recursivebond pricing parameters A and B. The estimates show that allelements of the vector λ̃0 governing the unconditional meanof the market prices of risk are large in absolute terms. Thissuggests that risk premia are characterized by an importantconstant component. However, the standard errors implied by thebootstrap algorithm are relatively large, so inference on the basisof individual estimates should be exercised with caution. A similarremark applies to the estimates of λ̃1 which govern the dynamiccomponent of themodel-implied risk premia.While there are clearsigns for time variation in the market prices of risk, only few of theestimated individual coefficients are statistically significant.

As has been noted in previous studies, it is difficult to pindown the market price of risk parameters in affine term structure

Table 6In-sample fit: Observed and model-implied yields

y(1) y(3) y(6) y(9) y(12) y(24) y(36) y(48) y(60) y(84) y(120)

Panel A: Mean

y(n) 5.22 5.47 5.62 5.74 5.89 6.27 6.55 6.78 6.90 7.14 7.27ŷ(n) 5.22 5.47 5.61 5.76 5.88 6.27 6.56 6.77 6.91 7.14 7.26|y(n)t − ŷ

(n)t | 0.00 0.14 0.19 0.24 0.29 0.41 0.46 0.50 0.51 0.56 0.58

Panel B: Standard deviation

y(n) 2.11 2.20 2.25 2.29 2.32 2.33 2.27 2.24 2.21 2.14 2.06ŷ(n) 2.12 2.19 2.25 2.28 2.29 2.27 2.21 2.16 2.12 2.04 1.92|y(n)t − ŷ

(n)t | 0.00 0.19 0.25 0.31 0.37 0.50 0.57 0.63 0.65 0.72 0.73

This table summarizes empirical means and standard deviations of observed and fitted yields. Yields are reported in percentage terms. The first and second row in eachpanel report the respective moment of observed yields and fitted values implied by the No-Arbitrage FAVAR model while in the third row the mean and standard deviationof absolute fitting errors are reported, respectively.


Fig. 2. Observed and model-implied yields. This figure provides plots of observed and model-implied time series for four selected interest rates, the 6-month yield, the12-month yield and the 3-and 10-year yields.

models.8 The lack of statistical significance of individual elementsof λ̃0 or λ̃1 found here is therefore not necessarily a sign ofpoor model fit. Yet, economic reasoning based on the significanceof individual parameters governing the state prices of risk isunwarranted. Instead, in order to visualize the relation betweenrisk premia and the model factors, Fig. 3 provides a plot of model-implied term premia for the 1-year and the 5-year yield. Asindicated by these plots, term premia at the short end of the yieldcurve are inversely related to the first macro factor which is itselfhighly correlated with output variables. By contrast, premia forlonger yields are more closely related to the second factor whichis strongly correlated with inflation indicators.

Fig. 4 provides a plot of the loadings bn of the yields ontothe contemporaneous observations of the model factors. Thesigns of these loadings are consistent with those obtained fromregressing yields onto the model factors without imposing no-arbitrage restrictions, summarized in Table 5. By construction ofmy arbitrage-free model, the loading of the 1-month yield ontothe short rate factor equals unity and those for the macro factorsare zero. However, the impact of the short rate on longer yieldssharply decreases with maturity. Hence, movements in the short-term interest rate only have a relatively small impact on long-term interest rates. Instead, these are more strongly driven by the

8 See e.g. Ang and Piazzesi (2003) or Hördahl et al. (2006).

Fig. 3. Risk premia dynamics. This figure provides a plot of the term premia forthe 1-year and 5-year yield as implied by the No-Arbitrage FAVAR model. Forcomparison, they are related to the first and second model factor, respectively.

macroeconomic factors. Most importantly, the first factor has anequally strong impact on yields of medium and longer maturities.


Table 7Parameter estimates for no-arbitrage FAVAR model

Φ1 Φ2

F1 0.977 −0.057 −0.107 −0.103 0.011 0.244 −0.143 0.013 0.143 0.044(0.096) (0.109) (0.118) (0.064) (0.061) (0.140) (0.180) (0.165) (0.088) (0.079)

F2 0.196 1.357 0.174 0.038 0.028 −0.055 −0.387 −0.306 0.005 0.086(0.064) (0.073) (0.079) (0.043) (0.041) (0.094) (0.121) (0.111) (0.059) (0.053)

F3 −0.160 0.098 0.945 −0.042 −0.043 0.112 −0.340 −0.014 0.072 0.012(0.072) (0.082) (0.088) (0.048) (0.046) (0.105) (0.135) (0.124) (0.066) (0.060)

F4 −0.102 −0.172 0.170 1.007 −0.071 −0.068 0.336 0.051 −0.192 −0.044(0.123) (0.140) (0.151) (0.082) (0.079) (0.179) (0.231) (0.212) (0.112) (0.102)

y(1) 0.140 0.086 −0.045 −0.106 0.860 0.163 −0.057 −0.198 0.147 −0.042(0.100) (0.113) (0.123) (0.066) (0.064) (0.146) (0.188) (0.173) (0.091) (0.083)

Φ3 Φ4

F1 −0.621 0.057 −0.006 −0.089 0.045 0.315 0.079 0.145 0.072 −0.102(0.139) (0.178) (0.165) (0.088) (0.079) (0.107) (0.107) (0.110) (0.061) (0.060)

F2 −0.171 −0.049 0.257 −0.028 −0.071 0.084 0.045 −0.117 −0.016 −0.040(0.094) (0.120) (0.111) (0.059) (0.053) (0.072) (0.072) (0.074) (0.041) (0.040)

F3 0.013 0.314 −0.350 −0.012 0.039 −0.087 −0.040 0.334 0.041 −0.006(0.104) (0.134) (0.124) (0.066) (0.059) (0.081) (0.081) (0.082) (0.046) (0.045)

F4 0.347 −0.358 −0.111 −0.259 0.091 −0.016 0.165 −0.030 0.293 0.040(0.178) (0.228) (0.212) (0.113) (0.101) (0.138) (0.138) (0.141) (0.078) (0.077)

y(1) −0.124 0.135 0.293 −0.001 −0.060 −0.022 −0.045 −0.082 −0.005 0.187(0.145) (0.186) (0.172) (0.092) (0.082) (0.112) (0.112) (0.114) (0.064) (0.063)

Ω µ

F1 0.100 0.013(0.009) (0.084)

F2 −0.020 0.045 −0.003(0.005) (0.004) (0.057)

F3 0.054 −0.013 0.057 −0.036(0.006) (0.003) (0.005) (0.063)

F4 −0.072 −0.016 −0.044 0.165 −0.091(0.010) (0.006) (0.007) (0.015) (0.108)

y(1) 0.006 −0.003 −0.005 −0.018 0.109 0.246(0.007) (0.005) (0.005) (0.009) (0.010) (0.088)

The market prices of risk specification is λt = λ0 + λ1Zt

λ̃0 λ̃1

−26.366 0.631 0.215 0.288 3.386 −0.084(29.514) (1.806) (0.920) (1.325) (1.416) (0.544)−10.220 4.673 0.338 0.870 −1.310 0.025(139.011) (5.064) (3.363) (3.461) (3.302) (1.772)−88.111 −0.635 −0.982 −0.164 −4.116 0.304(51.118) (2.600) (1.559) (1.580) (1.675) (0.771)−76.189 −1.612 −1.775 0.256 0.405 0.254(48.143) (2.119) (1.184) (1.481) (1.464) (0.652)−18.664 −0.208 −0.648 −0.044 −0.800 −0.246(12.770) (0.795) (0.658) (0.626) (0.604) (0.411)

This table provides estimates of the parameters of the FAVAR model obtained using the full sample information, i.e. from estimating the model over the 1983:01–2003:09period. The state dynamics are given by Zt = µ + Φ1Zt−1 + · · · Φ4Zt−4 + ωt , where E[ωtω′t ] = Ω . The states (F1 . . . F4) of the model have been extracted from the largepanel of macro time series using principal components methods.

Interestingly, shocks to the third macro factor appear to have anegative effect on yields of very short maturity and an increasinglystrong positive impact on medium-term and long-term rates. Thisindicates that negative shocks to the third macro factor imply aflattening of the yield curve which is commonly associated withan upcoming recession.

4.5. How are the macro factors related to the components of the yieldcurve?

In traditional term structure analysis, the yield curve is oftendecomposed into three factors which together explain almost allof the cross-sectional variation of interest rates. According to theirimpact on the shape of the term structure, these componentsare commonly labeled level, slope, and curvature. Since the No-Arbitrage FAVARmodel has been shown to explain yields relativelywell in-sample, it is interesting to relate the macro factors usedin the model to the level, slope, and curvature components ofthe yield curve. In this section, I thus regress estimates of the

latent yield factors onto the macro factors and the short rate. Theyield factors are computed as the first three principal componentsof the interest rates used to estimate the term structure model.Consistent with results reported in previous studies, these explainabout 90.8%, 6.4% and 1.6% of the total variance of all yields.

Table 8 summarizes the results of these regressions. The fourmacro factors and the short-term interest rate explain almostall of the variation in the yield level which captures the mostimportant source of common variation of interest rates. The maincontribution comes from the short rate and the first and thirdmacro factor which are correlated with output and inflation-related variables, respectively. Almost 80% of the variation in theslope of the yield curve is explained by the factors of the FAVARmodel. Again, the short rate as well as the first and third macrofactors are most strongly correlated with the slope. The short ratehas a strongly significant negative coefficient in the slope equationwhich is consistentwith the commonview that short rate increaseslead to a diminishing yield curve slope. Finally note that onlyabout 35% of the variation in the curvature of the yield curve are


Fig. 4. Implied yield loadings. This figure provides a plot of the yield loadings bnimplied by the No-Arbitrage FAVAR model. The coefficients can be interpreted asthe response of the n-month yield to a contemporary shock to the respective factor.

Table 8Regression of latent yield factors on the model factors

Level Slope Curvature

cst 0.23 1.65 −0.05[10.88] [9.40] [−0.11]

F1 0.04 0.13 −0.37[7.15] [4.18] [−3.91]

F2 0.03 0.10 −0.13[2.76] [1.48] [−0.96]

F3 0.04 0.30 0.02[5.89] [6.23] [0.30]

F4 0.02 0.14 −0.09[2.92] [2.44] [−1.26]

y(1) 0.07 −0.29 0.02[13.41] [−7.22] [0.33]

R̄2 0.95 0.77 0.35

This table summarizes the results obtained from a regression of level, slope, andcurvature yield factors onto the factors of the FAVAR model. Level, slope, andcurvature are computed as the first three principal components extracted from theyields used to estimate the term structuremodel. They explain 90.8%, 6.4% and 1.6%of the total variance of all yields, respectively. The sample period is 1983:01–2003:9.t-statistics are in brackets.

explained by the macro factors. Hence, variations in the relativesize of short, medium and long-term yields seem to be the leastrelated to changes in macroeconomic conditions.

5. Out-of-sample forecasts

The term structure model suggested in this paper is based onthe idea that the Federal Reserve uses a large set of conditioninginformation when setting short-term interest rates and that theFAVAR approach suggested by Bernanke et al. (2005) representsa useful way of capturing this information. While the modelcan in principle be employed to analyze e.g. the macroeconomicunderpinnings of yield curve dynamics, this paper focuses on theusefulness of the No-Arbitrage FAVAR model for predicting theterm structure of interest rates.

In the previous section, it has been shown that the modelprovides a reasonably good in-sample fit to US yield data. Inthis section, I study the forecast performance of the No-ArbitrageFAVAR model in a recursive out-of-sample prediction exercise.Before documenting the results of the forecasts, I briefly describehow they are computed for the different models studied in thispaper. I startwith theNo-Arbitrage FAVARmodel forwhichmodel-implied forecasts are obtained according to

ŷ(n)t+h|t = ân + b̂′

nẐt+h|t , (10)

where Z contains the contemporaneous and lagged observations ofthe four factors (F1, F2, F3, F4) explaining the bulk of variation inthe panel of monthly time series for the US and the 1-month yield,y(1). The four factors are re-estimated via principal componentseach period t a forecast is produced using data up to t . Thecoefficients ân and b̂n are computed according to equations (6) and(7), using as input the estimates µ̂, Φ̂, and Ω̂ obtained by runninga VAR on the states, as well as the estimates λ̂0 and λ̂1 that resultfrom minimizing the sum of squared fitting errors of the model.Forecasts Ẑt+h|t are obtained by iterating forward the FAVAR Eq.(2), i.e.

Ẑt+h|t = Φ̂hZt +

h−1∑i=0

Φ̂iµ̂. (11)

Note that the number of lags which enter the FAVAR equation arere-estimated every period a forecast is made on the basis of theHannan–Quinn criterionwith amaximum lag length of 12months.

5.1. The competitor models

I compare the model’s forecast performance to that of severalcompetitor models. In particular, these are a No-Arbitrage MacroVAR model, an unrestricted VAR on yield levels, two differentspecifications of the Nelson–Siegel (1987) three-factor modelrecently suggested by Diebold and Li (2006), an essentially affinelatent yield factor model following Duffee (2002), a simple AR(p)on yield levels, and the random walk. The Nelson–Siegel (1987)model is expected to be the most challenging competitor asDiebold and Li have shown that it outperforms a variety ofalternative yield forecasting models. In the following, I brieflysketch the individual competitor forecasting models.

5.1.1. No-Arbitrage macro VAR modelIn order to analyze whether the forecast performance of the

No-Arbitrage FAVAR model can be attributed to the large set ofconditioning information incorporated by the model, I compare itto a model that uses as state variables individual macroeconomicindicators. In particular, I compare it to a model that incorporatesas states the short rate and four macroeconomic variables whichare likely to contain information useful to explain yields.

Such a model has been suggested by Bernanke et al. (2004). Inaddition to the federal funds rate, these authors use the followingfour variables as states in their term structure model: a measureof the employment gap (payroll employment detrended by aHodrick–Prescott filter); inflation over the past year, as measuredby the deflator for personal consumption expenditures excludingfood and energy; expected inflation over the subsequent year,taken from the Blue Chip surveywhere inflation is defined in termsof the GDP deflator, and the year-ahead Eurodollar futures rate tocapture the expected path of monetary policy over the near-term.

Bernanke et al. (2004) report that their model explains theterm structure well over time. This result is confirmed byRudebusch et al. (2006b) who find that the model even providesa better fit to the cross-section of yields than originally describedby Bernanke et al. (2004). However, neither of both studiesinvestigates the forecast performance of the model. Here, I assessthe model’s ability to predict the yield curve out-of-sample ina recursive pseudo real-time setting. Precisely, I obtain yieldforecasts according to

ŷ(n)t+h|t = ân + b̂′

nẐVARt+h|t

where ZVAR = (Emp, π, π e, ED, y(1)) contains the cyclical compo-nent of payroll employment, PCE inflation, the Blue-Chip surveymeasure of expected inflation, the year-ahead Eurodollar futures


rate, and the 1-month yield. The coefficients ân and b̂n are ob-tained from Eqs. (6) and (7) and guarantee the absence of arbi-trage opportunities. Estimates of the model parameters based onthe entire sample 1983:01-2003:09 are provided in Appendix C.Forecasts ẐVARt+h|t are computed as in (11). As for the No-ArbitrageFAVARmodel, the lag order of the VAR is re-estimated on the basisof the Hannan–Quinn criterion with a maximum lag length of 12months every period a forecast is made. The No-Arbitrage MacroVAR model of Bernanke et al. (2004) is denoted ‘‘BRS’’ in the tablesbelow.

5.1.2. VAR(1) on yield levelsIn this model, forecasts of yields are obtained according to

ŷt+h|t = ĉ + Γ̂ yt ,

where ĉ and Γ̂ are estimated by regressing the vector yt ontoa constant and its h-months lag. This model is referred to as‘‘VARylds’’ in the results below.

5.1.3. Diebold–Li specification of the Nelson–Siegel modelIn a recent paper, Diebold and Li (2006) have suggested a dy-

namic version of the traditional Nelson–Siegel (1987) decompo-sition of yields and have shown that this model provides superioryield forecastswith respect to a number of benchmark approaches.According to this model, yields are decomposed into three factorswith loadings given by exponential functions of the time to matu-rity n and some shape parameter τ . Precisely, Diebold and Li sug-gest to obtain yield forecasts according to

ŷ(n)t+h|t = β̂1,t+h|t + β̂2,t+h|t

(1 − e−τn

τn

)+ β̂3,t+h|t

(1 − e−τn

τn− e−τn

)where

β̂t+h|t = ĉ + Γ̂ β̂t .

Diebold and Li (2006) obtain initial estimates of the factors β byregressing yields onto the loadings

(1, ( 1−e

−τn

τn ), (1−e−τn

τn − e−τn)

)for a fixed value of τ . They set τ = 0.0609 which is the valuethatmaximizes the curvature loading at thematurity of 30months.Diebold and Li consider two different specifications of their model,one where the factor dynamics are estimated by fitting AR(1)processes and another where the factors follow a VAR(1). In myapplication of their model, I report results for both specifications.These are denoted as ‘‘NS(AR)’’ and ‘‘NS(VAR)’’, respectively.

5.1.4. Essentially affine latent yield factor model A0(3)This is a traditional affine model where all the factors are latent

and have to be estimated from the yield data. I implement thepreferred essentially affine A0(3) specification of Duffee (2002)who has shown that this model provides superior out-of-sampleforecast results with respect to various other affine specifications.The specification of the market prices of risk is therefore similarto the No-Arbitrage FAVAR model. Within the A0(3) model, yieldforecasts are obtained as

ŷ(n)t+h|t = ân + b̂′

nẐA0(3)t+h|t

where ZA0(3) is composed of three latent yield factors, backed outfrom the yields using the method by Chen and Scott (1993). Inparticular, I assume that the 1-month, 1-year and 10-year yieldare observed without error. Moreover, the transition matrix Φin the state equation is assumed to be lower-triangular and thevariance-covariance matrix Ω to be an identity matrix so as to

ensure exact identification of the model (see Dai and Singleton(2000)). Notice that since the latent factors need to be backed outfrom the yields, estimation of the model takes considerably longerthan estimation of the No-Arbitrage FAVAR andMacro VARmodelswhere the parameters of the state equation are estimated in a firststage of the estimation via OLS.

5.1.5. AR(p) on yield levelsSimple autoregressive processes constitute another natural

benchmark for modeling the time variation of bond yields.Assuming that the yield of maturity n follows a p-th orderautoregression, its h-step ahead forecast is given by

ŷ(n)t+h|t = α̂0 + α̂1ŷ(n)t+h−1|t + · · · + α̂pŷ

(n)t+h−p|t

where ŷ(n)τ |t = y(n)τ for τ ≤ t.

In the implementation of thismodel, the lag order p is estimatedrecursively using the BIC information criterion.

5.1.6. Random walkFinally, many previous studies have suggested that the

evolution of interest rates might be well described by randomwalk processes. The random walk therefore remains a commonbenchmark for interest rate predictionmodels and is also used as acompetitor here. Assuming a randomwalkmodel for interest ratesimplies a simple no-change forecast of individual yields. Hence, inthis model the h-months ahead prediction of an n-maturity bondyield in period t is simply given by its time t observation:

ŷ(n)t+h|t = y(n)t .

5.2. Out-of-sample forecast results

The out-of-sample forecasts are carried out over the timeinterval 1994:01–2003:09. The forecast sample therefore covers aperiod of almost ten years. The affine models are first estimatedover the period 1983:01–1993:12 to obtain starting values for theparameters. All models are then re-estimated recursively usingdata from 1983:01 to the time that the forecast is made, beginningin 1994:01.

Table 9 summarizes the root mean squared errors obtainedfrom these forecasts. Three main observations can be made.First, the No-Arbitrage FAVAR model clearly outperforms the No-Arbitrage Macro VAR of Bernanke et al. (2004) for most maturitiesand especially in forecasts 6-months and 12-months ahead. Thisimplies strong support for the use of a broad macroeconomicinformation set when forecasting the yield curve based onmacroeconomic variables. Second, at the 1-month ahead horizon,the VAR(1) in yield levels and the AR(p) model outperform themacro-based FAVAR and VAR models for yields of all maturities,with the AR(p) being slightly superior for intermediate and longyields and the VARylds model performing best for the shortrate. Third and most importantly, however, the No-ArbitrageFAVARmodel dominates all considered benchmarkmodels in yieldforecasts 6-months and 12-months ahead. Indeed, as panels B andC of Table 9 document, the FAVAR model implies smaller out-of-sample root mean squared forecast errors than the benchmarkmodels except for the 10-year yield that is best predicted by theaffine latent yield factor model.

Interestingly, both specifications of the Nelson–Siegel modelconsidered in Diebold and Li (2006) are outperformed by theNo-Arbitrage FAVAR model. This result is striking since Dieboldand Li have documented their approach to be particularly goodat forecasting. This indicates that the combination of a largeinformation set, the rich dynamics of the FAVAR, and the parameterrestrictions implied by no-arbitrage together result in a model


Table 9Out-of-sample RMSEs—Forecast period 1994:01–2003:09

y(n) FAVAR BRS VARylds NS(VAR) NS(AR) A0(3) AR(p) RW

Panel A: 1-month ahead forecasts

1 0.534 0.331 0.249 0.262 0.275 0.681 0.305 0.3056 0.502 0.522 0.204 0.218 0.256 0.216 0.204 0.222

12 0.516 0.553 0.250 0.268 0.293 0.300 0.256 0.25936 0.630 0.485 0.308 0.313 0.312 0.386 0.299 0.30960 0.685 0.467 0.314 0.316 0.316 0.357 0.303 0.307

120 0.717 0.511 0.293 0.289 0.289 0.289 0.279 0.282

Panel B: 6-month ahead forecasts

1 0.601 0.854 0.779 0.745 0.838 1.189 0.860 0.8566 0.608 1.094 0.904 0.871 0.931 0.977 0.824 0.853

12 0.696 1.170 1.006 0.958 0.981 1.059 0.885 0.87636 0.756 1.073 1.021 0.958 0.922 0.962 0.897 0.87360 0.794 0.930 0.969 0.915 0.870 0.848 0.847 0.830

120 0.825 0.773 0.872 0.764 0.720 0.671 0.702 0.696

Panel C: 12-month ahead forecasts

1 0.919 1.539 1.366 1.448 1.357 1.741 1.397 1.3956 0.981 1.723 1.613 1.569 1.458 1.487 1.400 1.417

12 1.055 1.729 1.728 1.633 1.495 1.506 1.407 1.39136 1.066 1.473 1.599 1.504 1.349 1.264 1.253 1.23660 1.063 1.210 1.464 1.359 1.233 1.076 1.132 1.138

120 1.071 0.932 1.313 1.108 1.022 0.853 0.917 0.942

This table summarizes the root mean squared errors obtained from out-of-sample yield forecasts. The models have been estimated using data from 1983:01 until the periodwhen the forecast is made. The forecasting period is 1994:01–2003:09. ‘‘FAVAR’’ refers to the No-Arbitrage Factor-Augmented VAR model; ‘‘BRS’’ denotes the arbitrage-freeMacro VARmodel of Bernanke et al. (2004); ‘‘VARylds’’ refers to an unrestricted VAR(1) on yield levels; ‘‘NS(VAR)’’ and ‘‘NS(AR)’’ denote the Diebold–Li (2006) version of thethree-factor Nelson–Siegel model with VAR and AR factor dynamics, respectively; ‘‘A0(3)’’ refers to the essentially affine latent yield factor model of Duffee (2002); ‘‘AR(p)’’denotes an AR model where the lag order p is recursively estimated; ‘‘RW’’ refers to the random walk forecast.

Table 10RMSEs relative to random walk—Forecast period 1994:01–2003:09

y(n) FAVAR BRS VARylds NS(VAR) NS(AR) A0(3) AR(p)


1 1.751 1.085 0.816 0.859 0.900 2.232 1.0006 2.266 2.355 0.921 0.984 1.154 0.972 0.918

12 1.993 2.135 0.964 1.034 1.131 1.160 0.98736 2.039 1.571 0.996 1.013 1.011 1.250 0.96960 2.232 1.522 1.022 1.029 1.031 1.165 0.988

120 2.547 1.815 1.039 1.028 1.025 1.027 0.989


1 0.702 0.997 0.910 0.870 0.979 1.389 1.0046 0.712 1.283 1.059 1.022 1.092 1.145 0.966

12 0.795 1.336 1.148 1.094 1.119 1.209 1.01036 0.866 1.229 1.171 1.098 1.056 1.103 1.02860 0.956 1.121 1.167 1.102 1.048 1.022 1.021

120 1.186 1.112 1.254 1.099 1.035 0.964 1.010


1 0.659 1.103 0.979 1.038 0.973 1.249 1.0026 0.692 1.216 1.139 1.107 1.029 1.049 0.988

12 0.759 1.243 1.242 1.174 1.075 1.083 1.01236 0.863 1.192 1.293 1.217 1.091 1.023 1.01460 0.934 1.063 1.287 1.194 1.084 0.946 0.995

120 1.136 0.989 1.393 1.175 1.085 0.905 0.973

This table summarizes the root mean squared errors relative to those implied by the randomwalk. Themodels have been estimated using data from 1983:01 until the periodwhen the forecast is made. The forecasting period is 1994:01–2003:09. ‘‘FAVAR’’ refers to the No-Arbitrage Factor-Augmented VAR model; ‘‘BRS’’ denotes the arbitrage-freeMacro VARmodel of Bernanke et al. (2004); ‘‘VARylds’’ refers to an unrestricted VAR(1) on yield levels; ‘‘NS(VAR)’’ and ‘‘NS(AR)’’ denote the Diebold–Li (2006) version of thethree-factor Nelson–Siegel model with VAR and AR factor dynamics, respectively; ‘‘A0(3)’’ refers to the essentially affine latent yield factor model of Duffee (2002); ‘‘AR(p)’’denotes an AR model where the lag order p is recursively estimated; ‘‘RW’’ refers to the random walk forecast.

which is particularly useful for out-of-sample predictions. In thesubsample analysis carried out in the next section, I will analyzethis result in more detail.

Table 10 reports RMSEs of all considered models relative to therandomwalk forecast. These results show that the improvement interms of root mean squared forecast errors implied by the FAVARmodel is particularly pronounced for short and medium term ma-turities. At the one-month forecast horizon, all yield-basedmodelsoutperform the affine models based on macro variables. However,

at forecast horizons beyond one month, the No-Arbitrage FAVARmodel outperforms all othermodels formaturities fromonemonthto five years. Relative to the randomwalk, the suggestedmodel re-duces root mean squared forecast errors up to 35% at the short endof the yield curve and improves forecasts of medium-term yieldsup to 15%.While all considered competitor models outperform therandomwalk in 6-months and 12-months ahead forecasts only forsome maturities, the No-Arbitrage FAVAR model consistently out-performs the RandomWalk except for the 10-year yield.


Table 11White’s reality check test—Forecast period 1994:01–2003:09

y(n) BRS VARylds NS(VAR) NS(AR) A0(3) AR(p) RW


1 1.881 2.390 2.315 2.241 −1.915 2.055 2.0576 −0.201 2.274 2.212 2.019 2.222 2.277 2.195

12 −0.388 2.214 2.115 1.963 1.916 2.180 2.16436 1.798 3.298 3.266 3.266 2.719 3.349 3.29360 2.774 4.055 4.041 4.032 3.745 4.116 4.101

120 2.797 4.693 4.716 4.720 4.716 4.766 4.761


1 −3.966 −2.727 −2.062 −3.581 −11.050 −4.024 −3.9466 −8.916 −4.883 −4.268 −5.311 −6.266 −3.365 −3.943

12 −9.525 −5.788 −4.823 −5.148 −6.892 −3.337 −3.22336 −6.324 −5.235 −3.936 −3.073 −3.974 −2.774 −2.31460 −2.608 −3.524 −2.448 −1.489 −1.186 −1.260 −0.878

120 0.816 −1.051 0.756 1.546 2.246 1.693 1.862


1 −15.919 −10.630 −12.891 −10.170 −22.505 −11.589 −11.5046 −21.081 −17.180 −15.790 −12.063 −13.331 −10.608 −11.404

12 −19.634 −19.526 −16.494 −11.750 −12.519 −9.291 −9.24836 −11.096 −14.876 −12.092 −7.259 −5.390 −5.132 −4.82160 −3.814 −10.630 −7.755 −4.200 −0.770 −2.314 −2.214

120 2.625 −6.008 −1.121 0.915 4.040 2.502 2.348

This table summarizes ‘‘White’s Reality Check’’ test statistics based on a squared forecast error loss function. I choose the No-Arbitrage FAVARmodel as the benchmarkmodeland compare it bilaterally with the competitor models. Negative test statistics indicate that the average squared forecast loss of the FAVAR model is smaller than that of therespective competitor model. Bold figures indicate significance which is checked by comparing the average forecast loss differential with the 5% percentile of the empiricaldistribution of the loss differential series approximated by applying a block bootstrap with 1000 resamples and a smoothing parameter of 1/12.

One can formally assess whether the improvement of theFAVAR model over the benchmark models in terms of forecasterror is significant by applying White’s (2000) ‘‘reality check’’ test.This test uses bootstrap resamples of the forecast error series toderive the empirical distribution of the forecast loss differential ofa model with respect to some benchmark model. It can thus beemployed to evaluate the superior predictive ability of a modelas compared to one or more competitor models. Here, I testwhether the No-Arbitrage FAVAR model has superior predictiveaccuracy with respect to the seven considered competitors. Thetest statistics are reported in Table 11. Negative numbers indicatethat the average squared forecast loss of the No-Arbitrage FAVARmodel is smaller than that of the respective competitor modelwhile positive test statistics indicate the opposite. I perform 1000block-bootstrap resamples from the prediction error series tocompute the significance of the forecast improvement at the 5%level which are indicated by bold figures. As the results in panelsB and C of Table 11 show, the documented improvement in termsof root mean squared forecast errors is significant at the 5% levelfor all but very long maturities at forecast horizons of 6-monthsand 12-months ahead. This underscores the observation madeabove that the No-Arbitrage FAVAR model predicts interest ratesconsiderably better than all studied competitor models, includingthe Nelson–Siegel model and the A0(3) model.

5.3. Subsample analysis of forecast performance

The results documented in the previous section show that theNo-Arbitrage FAVAR model exhibits strong relative advantagesover a variety of benchmarkmodels which have been documentedpowerful tools in forecasting the yield curve. This result somewhatchallenges the recent findings of Diebold and Li (2006) andtherefore a closer look at the predictive ability of the differentmodels is warranted. In this section, I thus perform a subsampleanalysis of the out-of-sample prediction results. In particular, Ianalyze the relative performance of theNo-Arbitrage FAVARmodelwith respect to the Nelson–Siegel model over exactly the sampleperiod that has been studied by Diebold and Li (2006).

Table 12 provides the root mean squared forecast errors ofthe different models for the out-of-sample prediction period1994:01–2000:12. At the 1-month ahead horizon, both specifica-tions of the Nelson–Siegel model outperform the other modelsexcept for the AR(p) model and the random walk which predictmaturities from 6 months to 5-years better. The absolute size ofthe RMSEs is very similar to those documented by Diebold and Li(2006). For example, based on the NS(AR) model Diebold and Li re-port RMSEs of 0.236, 0.292, and 0.260 for the 1-year, 5-year and10-year yields at the 1-month ahead horizon whereas I find valuesof 0.249, 0.280, and 0.249, respectively, for the same maturities.The small deviations are likely due to differences in the choice ofdata and the set of maturities used to estimate themodels. Turningto the results for 6-months ahead predictions, the picture becomesless favorable for the Nelson–Siegel model. Only for the 1-monthyield, the VAR specification of the Nelson–Siegel model performsbest. In contrast, the No-Arbitrage FAVAR model outperforms allother models for the range of maturities between 6-months and5-years. Again, the absolute size of the RMSEs found here is verysimilar to those reported by Diebold and Li. For example, whilethese authors document RMSEs of 0.669, 0.777, and 0.721 for the1-year, 5-year and 10-year yields, I find values of 0.711, 0.764, and0.694, respectively. The results again change somewhat if one con-siders 12-months ahead predictions for the sample period studiedin Diebold and Li (2006). In this case, there appears to be a cleareradvantage of their preferred NS(AR) specification which outper-forms all other models except for the 10-year maturity.

To visualize these results, Figs. 5 to 7 show the actual yields andthose predicted by the No-Arbitrage FAVAR, the BRS, the NS(AR),and the A0(3) model for some selected maturities. Fig. 5 plots theoutcomes for the 1-month ahead forecast horizon. According tothis plot, the NS(AR) and the A0(3) model forecast the persistentmovements of yields quite well while the FAVAR model predictsmore variation than actual yields exhibit. In particular, themodel’spredictions appear to be particularly poor around turning points ofyield dynamics. Interestingly, the same observation applies to theBRS model. Hence, both models that are based on macroeconomicinformation tend to overstate the volatility of interest rates. Thisconfirms the relatively poor predictive ability of the two models


Table 12Out-of-sample RMSEs—Forecast Period 1994:01–2000:12



1 0.380 0.313 0.255 0.265 0.249 0.722 0.299 0.2976 0.357 0.479 0.194 0.186 0.215 0.209 0.180 0.192

12 0.395 0.559 0.242 0.238 0.249 0.280 0.236 0.23936 0.566 0.488 0.281 0.286 0.272 0.368 0.267 0.27760 0.683 0.468 0.290 0.289 0.280 0.343 0.276 0.275

120 0.804 0.555 0.270 0.256 0.249 0.254 0.254 0.253


1 0.625 0.893 0.696 0.509 0.532 1.140 0.683 0.6356 0.594 1.108 0.799 0.660 0.648 0.936 0.719 0.655

12 0.656 1.199 0.898 0.778 0.711 0.999 0.815 0.74236 0.650 1.142 0.947 0.877 0.747 0.938 0.895 0.83460 0.742 0.994 0.949 0.885 0.764 0.834 0.868 0.821

120 0.879 0.826 0.911 0.793 0.694 0.637 0.760 0.730


1 0.900 1.537 1.025 0.899 0.812 1.654 1.104 0.9456 0.910 1.687 1.179 1.002 0.908 1.430 1.177 0.977

12 0.952 1.702 1.268 1.078 0.932 1.414 1.205 1.01736 0.983 1.532 1.333 1.168 0.937 1.188 1.228 1.07860 1.055 1.284 1.331 1.179 0.979 1.007 1.176 1.072

120 1.160 1.005 1.333 1.089 0.941 0.775 1.028 0.985


Table 13Out-of-sample RMSEs—Forecast period 2000:01–2003:09



1 0.762 0.385 0.300 0.296 0.349 0.648 0.371 0.3666 0.699 0.566 0.214 0.256 0.316 0.228 0.226 0.257

12 0.674 0.493 0.250 0.297 0.348 0.326 0.271 0.28136 0.726 0.420 0.336 0.341 0.359 0.396 0.329 0.34260 0.660 0.406 0.339 0.344 0.360 0.366 0.328 0.342

120 0.488 0.362 0.310 0.330 0.336 0.325 0.306 0.312


1 0.581 0.761 0.896 1.027 1.231 1.391 1.120 1.1656 0.617 1.058 1.038 1.148 1.298 1.085 0.978 1.118

12 0.735 1.090 1.153 1.213 1.327 1.169 0.988 1.07836 0.879 0.885 1.150 1.095 1.158 0.972 0.887 0.95660 0.830 0.745 1.019 0.969 1.020 0.841 0.797 0.868

120 0.656 0.586 0.798 0.716 0.759 0.709 0.562 0.634


1 0.939 1.571 1.896 2.191 2.079 1.927 1.911 2.0526 1.116 1.848 2.297 2.382 2.221 1.654 1.823 2.108

12 1.257 1.829 2.459 2.461 2.288 1.749 1.805 2.03036 1.224 1.385 2.085 2.084 1.974 1.471 1.372 1.60160 1.069 1.066 1.738 1.711 1.660 1.252 1.111 1.329

120 0.842 0.760 1.275 1.175 1.185 1.022 0.716 0.891


at very short forecast horizons documented above. Yet, at the6-months ahead forecast horizon the picture looks strikinglydifferent. In particular, as Fig. 6 shows, the No-Arbitrage FAVARmodel predicts the surge of yields in 1999 and 2000 quite well.More impressively, it forecasts the strong decline of yields startingin late 2000 very precisely. By contrast, both the NS(AR) and theA0(3) models miss the particular dynamics in this episode by afewmonths. The affine macro VARmodel of Bernanke et al. (2004)forecasts the strong decline of interest rates somewhat earlier

than these two models, but overstates short and medium termmaturities at the end of the sample. Although less pronounced,a similar pattern can be seen for the 12-months ahead forecasts,provided in Fig. 7.

Altogether, these results show that the No-Arbitrage FAVARmodel performs particularly well compared to yield-based predic-tionmodels in periodswhen interest rates exhibit strong variation.To provide amore quantitative assessment of this finding, Table 13displays the root mean squared forecast errors of the different


Fig. 5. Observed and predicted yields—1 month ahead. This figure provides plots of the observed and 1-month ahead predicted time series for the 1-month, the 12-month,the 3- and 10-year maturities. The observed yields are plotted by solid lines, whereas dashed, dash-dotted, and dotted lines correspond to predictions of the No-ArbitrageFAVAR model, the NS(AR) model, and the A0(3) model, respectively.

models for the subperiod 2000:01–2003:09. As can be seen fromthe plots above, this period was characterized by an initial surgeof yields which was then followed by a sharp and persistent de-cline of interest rates of all maturities. The results of Table 13 showthat over this particular sample period, the No-Arbitrage FAVARmodel strongly outperforms all competitormodels at forecast hori-zons 6-months and 12-months ahead for maturities up to 3 years.More precisely, the reduction in RMSEs relative to the randomwalkamounts to a striking 50% for very short maturities. Over the samesubperiod, the 5-year and 10-year yields are best predicted by theBRS and AR(p)-models, respectively.

In sum, the results of the subsample analysis show that someof the strong forecast performance of the Nelson–Siegel modeldocumented by Diebold and Li may be due to their choice offorecast period. In addition, the superior predictive ability of themodel partly vanishes when confronted with the No-ArbitrageFAVAR model which strongly outperforms all benchmark modelsin periods when interest rates move a lot.

6. Conclusion

This paper presents a model of the term structure based onthe idea that the central bank uses a large set of conditioninginformation when setting the short term interest rate and that thisinformation can be summarized by a few factors extracted from

a large panel of macroeconomic time series. Precisely, the Factor-Augmented VAR (FAVAR) approach suggested by Bernanke et al.(2005) is used to model the dynamics of the short-term interestrate. Given this dynamic characterization of the short rate, theterm structure is then built up using restrictions implied by no-arbitrage. This setup is labeled a ‘‘No-Arbitrage Factor-AugmentedVAR’’ approach. In contrast to most previously proposed macro-finance models of the term structure, the model suggested in thispaper does not contain latent yield factors, but is entirely builtupon macroeconomic information.

Fitting the model to US data, I document that it explainsthe dynamics of yields quite well. This underlines that mostof the variation of interest rates is captured by macroeconomicvariables. Most importantly, I find that the No-Arbitrage FAVARmodel exhibits a strikingly good ability to predict the yield curveout-of-sample. In particular at intermediate and long forecasthorizons, the model outperforms various benchmarks includingthe essentially affine three factor model of Duffee (2002) and thedynamic variant of the Nelson–Siegel model that Diebold and Li(2006) have recently suggested as a predictionmodel. A subsampleanalysis of the forecast results documents that the No-ArbitrageFAVAR model performs particularly well in periods when interestrates exhibit pronounced dynamics.

Based on the findings of the paper, there are a number ofinteresting directions for future research. First, while this paperhas focused on the predictive ability of the No-Arbitrage FAVAR


Fig. 6. Observed and predicted yields—6months ahead. This figure provides plots of the observed and 6-months ahead predicted time series for the 1-month, the 12-month,the 3- and 10-year maturities. The observed yields are plotted by solid lines, whereas dashed, dash-dotted, and dotted lines correspond to predictions of the No-ArbitrageFAVAR model, the NS(AR) model, and the A0(3) model, respectively.

approach, the model can also be used for structural economicanalysis. For example, it would be interesting to identify monetarypolicy shocks as in Bernanke et al. (2005) and study their impacton the yield curve. Second, based on estimates of term premia,one could use the model to analyze the risk-adjusted expectationsof future monetary policy conditional on all macro informationavailable. Finally, estimating themodel using a one-step likelihoodbased Bayesian approach, one could easily add latent yield factorsand assess to what extent these enhance the explanatory andpredictive power of the model.

Acknowledgements

I would like to thank two anonymous referees, the editorArnold Zellner, Theofanis Archontakis, Jean Boivin, Claus Brand,Albert Lee Chun, Sandra Eickmeier, Lars Hansen, PhilippHartmann,Peter Hördahl, Manfred Kremer, Monika Piazzesi, Diego RodriguezPalenzuela, Glenn Rudebusch, Oreste Tristani, Harald Uhlig, DavidVestin, Thomas Werner, Jonathan Wright and various seminarparticipants for helpful comments. I am further grateful to RobertBliss, Lucrezia Reichlin, and Brian Sack for sharing their data withme. Parts of this work were done when the author was visitingthe ECB and the University of Pennsylvania. Financial supportby the German National Academic Foundation, the DeutscheForschungsgemeinschaft through the ‘‘SFB 649’’, and the Fritz-Thyssen Foundation is gratefully acknowledged. All remainingerrors are the author’s responsibility.

Appendix A. Derivation of the bond pricing parameters

The absence of arbitrage between bonds of different maturityimplies the existence of the stochastic discount factorM such that

P (n)t = Et [Mt+1 P(n−1)t+1 ],

i.e. the price of a n-months to maturity bond in month t mustequal the expected discounted price of an (n − 1)-months tomaturity bond inmonth (t+1). Following Ang and Piazzesi (2003),the derivation of the recursive bond pricing parameters starts byassuming that the nominal pricing kernelM takes the form

Mt+1 = exp(

−rt −12λ′tΩλt − λ

′

tωt+1

)and by guessing that bond prices P are exponentially affine in thestate variables Z , i.e.

P (n)t = exp(An + B′

nZt).

Plugging the above expressions for P and M into the firstrelation, one obtains

P (n)t = Et [Mt+1 P(n−1)t+1 ]

= Et

[exp

(−rt −

12λ′tΩλt − λ

′

tωt+1

)exp(An−1 + B′n−1Zt+1)

]= exp

(−rt −

12λ′tΩλt + An−1

)


Fig. 7. Observed and predicted yields—12 months ahead. This figure provides plots of the observed and 12-months ahead predicted time series for the 1-month, the12-month, the 3- and 10-year maturities. The observed yields are plotted by solid lines, whereas dashed, dash-dotted, and dotted lines correspond to predictions of theNo-Arbitrage FAVAR model, the NS(AR) model, and the A0(3) model, respectively.

× Et[exp(−λ′tωt+1 + B

′

n−1(µ + ΦZt + ωt+1))]

= exp(

−rt −12λ′tΩλt + An−1 + B

′

n−1µ + B′

n−1ΦZt

)× Et

[exp((−λ′t + B

′

n−1)ωt+1)].

Since the innovations ω of the state variable process areassumedGaussianwith variance-covariancematrixΩ, it is obviousthat

ln Et[exp((−λ′t + B

′

n−1)ωt+1)]

= Et[ln(exp((−λ′t + B

′

n−1)ωt+1))]

+12Vart

(ln(exp((−λ′t + B

′

n−1)ωt+1)))

=12

[λ′tΩλt − 2B

′

n−1Ωλt + B′

n−1ΩBn−1]

=12λ′tΩλt − B

′

n−1Ωλt +12B′n−1ΩBn−1.

Hence, Et[exp((−λ′t + B

′

n−1)ωt+1)]

= exp( 12λ′tΩλt − B

′

n−1Ωλt

+12B

′

n−1ΩBn−1) and thus

P (n)t = exp(

−rt −12λ′tΩλt + An−1 + B

′

n−1µ + B′

n−1ΦZt + · · ·

+12λ′tΩλt − B

′

n−1Ωλt +12B′n−1ΩBn−1

).

Using the relations rt = δ′Zt and λt = λ0 + λ1Zt , and matchingcoefficients finally yields

P (n)t = exp(An + B′

nZt),where

An = An−1 + B′n−1(µ − Ωλ0) +12B′n−1ΩBn−1,

and B′n = B′

n−1(Φ − Ωλ1) − δ′.

These are the recursive equations of the pricing parameters statedin (6) and (7).

Appendix B. Computation of standard errors by Monte Carlo

The two-step approach used to estimate the No-ArbitrageFAVARmodel implies a potential errors-in-variables bias since theestimation of themarket price of risk parameters takes as given theestimated evolution of the states. To adjust for this bias, I computestandard errors for λ0 and λ1 using the following Monte Carloprocedure.

From the estimation of the model, I save the model-impliedpricing errors and state innovations, respectively. Using thestationary block bootstrap of Politis and Romano (1994a,b),9 I then

9 This algorithm delivers blocks of time indexes that are of random length anddistributed according to the Geometric distribution with mean block length equalto 1/q where q is a smoothing parameter to be chosen. In the implementation ofthis algorithm, I set q = 1/12 which implies a mean block length of 12 months.


Table 14Parameter estimates for the BRS model: State dynamics

Φ1 Φ2

Emp 0.842 −0.146 0.370 −0.188 −0.362 0.324 0.369 −1.703 −0.078 −0.024(0.151) (0.417) (0.820) (0.161) (0.232) (0.116) (0.536) (1.090) (0.193) (0.318)

π 0.009 0.878 0.064 0.026 −0.044 −0.030 −0.090 −0.067 −0.012 0.025(0.010) (0.132) (0.144) (0.021) (0.036) (0.011) (0.086) (0.182) (0.038) (0.033)

π e −0.003 0.054 1.030 0.057 0.042 −0.006 0.015 −0.115 −0.068 −0.048(0.004) (0.033) (0.078) (0.014) (0.023) (0.006) (0.046) (0.110) (0.025) (0.032)

ED −0.045 0.326 −0.296 1.106 0.104 −0.000 −0.026 0.463 −0.299 −0.063(0.019) (0.176) (0.290) (0.059) (0.107) (0.027) (0.178) (0.396) (0.099) (0.142)

y(1) −0.024 0.074 −0.073 0.243 0.814 −0.025 −0.120 −0.244 −0.144 −0.005(0.020) (0.111) (0.369) (0.052) (0.093) (0.024) (0.132) (0.500) (0.087) (0.104)

Φ3 Φ4

Emp 0.031 0.332 1.899 −0.239 0.063 −0.261 −0.209 −0.760 0.424 0.224(0.073) (0.596) (1.160) (0.388) (0.288) (0.057) (0.549) (0.807) (0.273) (0.206)

π 0.010 0.152 0.063 −0.029 −0.037 0.006 0.050 −0.056 0.025 0.036(0.011) (0.079) (0.179) (0.043) (0.040) (0.009) (0.075) (0.105) (0.027) (0.036)

π e 0.002 −0.007 −0.024 0.030 0.033 0.007 0.014 0.000 −0.010 −0.027(0.007) (0.042) (0.103) (0.021) (0.023) (0.004) (0.032) (0.062) (0.012) (0.016)

ED 0.038 −0.237 −0.028 0.134 −0.025 0.010 −0.063 −0.034 −0.077 0.076(0.032) (0.151) (0.486) (0.079) (0.084) (0.020) (0.163) (0.331) (0.067) (0.073)

y(1) 0.021 0.139 0.553 0.020 −0.087 0.024 −0.003 −0.350 −0.027 0.160(0.023) (0.129) (0.362) (0.081) (0.078) (0.017) (0.129) (0.339) (0.052) (0.060)

Ω µ

Emp 1.485 0.552(0.505) (0.445)

π 0.003 0.028 0.034(0.013) (0.006) (0.069)

π e −0.005 0.002 0.006 0.045(0.007) (0.001) (0.001) (0.025)

ED −0.037 0.006 0.003 0.159 0.078(0.033) (0.004) (0.002) (0.017) (0.145)

y(1) −0.016 −0.000 0.004 0.038 0.108 0.054(0.027) (0.003) (0.002) (0.010) (0.022) (0.139)

This table provides estimates of the parameters of the BRS model obtained using the full sample information, i.e. from estimating the model over the 1983:01–2003:09period. The state dynamics are given by Zt = µ + Φ1Zt−1 + · · · Φ4Zt−4 + ωt , where E[ωtω′t ] = Ω . The states of the BRS model are given by the cyclical component ofpayroll employment (Emp), PCE inflation (π ), the Blue-Chip survey measure of expected inflation (π e), the year-ahead Eurodollar futures rate (ED), and the 1-month yield(y(1)).

Table 15Parameter estimates for the BRS model: Market prices of risk

λ̃0 λ̃1

−18.781 −0.028 0.479 −0.670 −0.083 −0.004(30.717) (0.220) (0.921) (1.063) (0.443) (0.368)−94.238 −0.310 0.456 1.059 −1.258 0.761(120.554) (1.341) (4.178) (4.491) (2.651) (2.010)−64.335 0.171 2.701 1.381 0.408 0.532(288.949) (2.351) (7.274) (10.629) (4.186) (2.960)−42.076 0.054 −0.162 0.101 −0.610 0.459(38.405) (0.266) (1.235) (1.386) (0.620) (0.443)15.458 −0.034 0.783 −1.501 0.181 −0.274(21.464) (0.247) (1.361) (2.055) (0.810) (0.716)

Date post:	21-Oct-2020
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